Theorem 2.4 (Multiplicity-free Fischer decomposition, [1]}) 
Under the joint action of \(\mathfrak {so}(m) \times \mathfrak {sl}(2)\) , we obtain the following multiplicity-free decomposition of the polynomial space \(\mathcal {P}(\mathbb {R}^m,\) :
\( \mathcal {P}(\mathbb {R}^{m}, \cong \bigoplus _{k=0}^{\infty }\mathcal {H}_k(\mathbb {R}^m,\otimes \mathbb {V}_{k+\frac{m}{2}}^{\infty },\) 
